3000w / 14.4v = 208 amps.
208 amps / .85 % (amp efficiency) = 245 amps is what you should expect your amp to draw. 190 seems underrated.
100 amps for the factory vehicle is probably a little oversized on what you actually use, but that's your fuel injection, computer, headlights, turn signals, brake lights, gauges, your two biggest are electric fans and Air conditioner.
320a sounds about right size for your alt, if that's what your alt will put out. If you haven't bought the alt yet, you want to know two things. Amps at (or near) idle, and what rpm for full output. If a manufacturer can't give you both, then keep looking. A cheap 320 alt will do less at idle than your factory 100, and won't do max output until over 3K rpms, (which you will never be cruising at). A quality alt will do more than half at idle, and full somewhere around 2K. They will also be able to give you an actual curve of your actual alternator. You should know those numbers, and where your cruising rpm is, before buying.
You're clearly going to need an alt to keep that going. You will probably need a battery too, but if you do the alt, and your lights dim as it hits, (and you have done the big 3, and your electrical is good), then your battery is too small, (too small of an output). If you crank it up, and your voltage drops after 30minutes to an hour, then your alt is too small.
A battery is a storage bank. If you only listen to your music when your engine is running, (and your alt is charging), then you only need enough to keep it going. If you play your music with the engine off, and need it to play for a longer time, then you would need another battery so you have a reserve / larger storage bank. You probably need a new battery, but what you don't want is to buy an alternator, and it's just too small, and then you need another one. There is no reason you can't go bigger if that's an option, but your numbers sound like they work if the RPMs will keep that alt charging at the rate you need it to.
